# Hest

##### Spherical Contact Distribution Function

Estimates the spherical contact distribution function of a random set.

- Keywords
- spatial, nonparametric

##### Usage

`Hest(X, ...)`

##### Arguments

- X
- The observed random set.
An object of class
`"ppp"`

,`"psp"`

or`"owin"`

. - ...
- Arguments passed to
`as.mask`

to control the discretisation.

##### Details

The spherical contact distribution function
of a stationary random set $X$
is the cumulative distribution function $H$ of the distance
from a fixed point in space to the nearest point of $X$,
given that the point lies outside $X$.
That is, $H(r)$ equals
the probability that `X`

lies closer than $r$ units away
from the fixed point $x$, given that `X`

does not cover $x$.

For a point process, the spherical contact distribution function
is the same as the empty space function $F$ discussed
in `Fest`

.

For `Hest`

, the argument `X`

may be a point pattern
(object of class `"ppp"`

), a line segment pattern
(object of class `"psp"`

) or a window (object of class
`"owin"`

). It is assumed to be a realisation of a stationary
random set.

The algorithm first calls `distmap`

to compute the
distance transform of `X`

, then computes the Kaplan-Meier
and reduced-sample estimates of the cumulative distribution
following Hansen et al (1999).

##### Value

- An object of class
`"fv"`

, see`fv.object`

, which can be plotted directly using`plot.fv`

.Essentially a data frame containing five columns:

r the values of the argument $r$ at which the function $H(r)$ has been estimated rs the ``reduced sample'' or ``border correction'' estimator of $H(r)$ km the spatial Kaplan-Meier estimator of $H(r)$ hazard the hazard rate $\lambda(r)$ of $H(r)$ by the spatial Kaplan-Meier method raw the uncorrected estimate of $H(r)$, i.e. the empirical distribution of the distance from a fixed point in the window to the nearest point of `X`

##### References

Baddeley, A.J. Spatial sampling and censoring.
In O.E. Barndorff-Nielsen, W.S. Kendall and
M.N.M. van Lieshout (eds)
*Stochastic Geometry: Likelihood and Computation*.
Chapman and Hall, 1998.
Chapter 2, pages 37-78.
Baddeley, A.J. and Gill, R.D.
The empty space hazard of a spatial pattern.
Research Report 1994/3, Department of Mathematics,
University of Western Australia, May 1994.

Hansen, M.B., Baddeley, A.J. and Gill, R.D.
First contact distributions for spatial patterns:
regularity and estimation.
*Advances in Applied Probability* **31** (1999) 15-33.

Ripley, B.D. *Statistical inference for spatial processes*.
Cambridge University Press, 1988.

Stoyan, D, Kendall, W.S. and Mecke, J.
*Stochastic geometry and its applications*.
2nd edition. Springer Verlag, 1995.

##### See Also

##### Examples

```
X <- runifpoint(42)
H <- Hest(X)
Y <- rpoisline(10)
H <- Hest(Y)
data(heather)
H <- Hest(heather$coarse)
```

*Documentation reproduced from package spatstat, version 1.16-3, License: GPL (>= 2)*